1. LING 364: Introduction to Formal Semantics Lecture 10 February 14th

2. Administrivia Reminder –Homework 2 due tonight we did Exercises 1 through 3 in the lab class last Thursday need more help? see me after class today...

3. Administrivia Thursday –(3:30pm – 4:45pm) Computer Lab Class meet in Social Sciences 224 instead of here

4. Last Time Grammar Rule Recursion Recursion: –A phrase may contain embedded inside another instance of the same phrase Examples: –sentence with a relative clause [ Sbar [ S I saw [ NP the man [ Sbar who [ S attacked me]]]]] –possessive NPs [ NP [ NP [ NP John]’s mother]’s cat]

5. Last Time Grammar Rule Recursion (Fixed) Prolog Computation Rule: –always pick the first-mentioned matching grammar rule to try each time we expand a non-terminal General Rule for writing recursive rules: –put recursive case last –i.e. place non-recursive rules for a non-terminal ahead of the recursive ones DCG rules for Possessive NPs: –np --> np, [‘‘‘s‘], n. –n --> [mother]. –n --> [cat]. –np --> [john]. move recursive rule to the end avoid Infinite Loop in Prolog ERROR: out of local stack.

6. Last Time Chapter 3: More about Predicates Lambda Calculus vs. Prolog notation –easy to understand as just “syntactic sugar” i.e. just an equivalent way of expressing what we’ve been using Prolog for –every logic variable, e.g. X, must be “quantified” using lambda, e.g. λx. –result is a slightly more complicated-looking notation Example: –PhraseLambda CalculusProlog notation –barksλx.x barks barks(X). –Shelby barks[λx.x barks](Shelby) barks(X), X = shelby. Example (Quiz 3) transitive predicate : –PhraseLambda CalculusProlog notation –likes λy.[λx.x likes y] likes(X,Y). –likes Mary [λy.[λx.x likes y]](Mary) likes(X,Y), Y = mary.

7. Today’s Topic “The Lambda Calculus Lecture” –Getting comfortable with Lambda Calculus see it as another way of stating what we have been doing already using Prolog notation –do lots of examples

8. More on the Lambda Calculus Lambda Calculus vs. Prolog notation Example (Quiz 3) transitive predicate: –PhraseLambda CalculusProlog notation –likesλy.[λx.x likes y] likes(X,Y). –likes Mary [λy.[λx.x likes y]](Mary) likes(X,Y), Y = mary. – λx.x likes Mary likes(X,mary). –John likes Mary [λx.x likes Mary](John) likes(X,mary), X = john. – John likes Mary likes(john,mary).

9. More on the Lambda Calculus How to do variable substitution –Official Name: Beta (β)-reduction –ExampleExpression –likes [λy.[λx.x likes y]] –likes Mary [λy.[λx.x likes y]](Mary) –means (basically): –(1) delete the outer layer, i.e. remove [λy. ☐ ](Mary) part, and –(2) keep the ☐ part, and –(3) replace all occurrences of the deleted lambda variable y in ☐ with Mary [λy.[λx.x likes y]](Mary) [λx.x likes y][λy. ](Mary) [λx.x likes Mary]

10. More on the Lambda Calculus Note: nesting order of λy and λx matters why: λy.[λx.x likes y] λx.[λy.x likes y] here: lambda expression quantifier for the object must be outside because of phrase structure hierarchy Example: PhraseLambda Calculus likesλy.[λx.x likes y] likes Mary [λy.[λx.x likes y]](Mary) λx.x likes Mary John likes Mary [λx.x likes Mary](John) John likes Mary sentence np vp v John Mary likes marylikes(X,Y) likes(X,mary)john likes(john,mary) sentence np vp v John Mary likes Mary λy.[λx.x likes y] λx.x likes MaryJohn John likes Mary Prolog notation Lambda Calculus

11. More on the Lambda Calculus 3.3 Relative Clauses –(7) Hannibal is [who Shelby saw] semantics of relative clause [who Shelby saw]: –who Shelby saw is a bit like a sentence (proposition) who 1 Shelby saw e 1 wh-movement of who 1 leaving a trace e 1 Shelby saw whounderlying structure Prolog style: saw(shelby,who). saw(shelby,X). (using a logic variable for who) lambda calculus style: λx.Shelby saw x(straight translation from Prolog)

12. More on the Lambda Calculus We’re going to compare: –(7) Hannibal is [who Shelby saw] –(7’) Hannibal is happy Consider the semantics of (7’) sentence np vp v Hannibal happy is happy(X) hannibal happy(hannibal) Prolog notation cf. Homework 2 John is a student student(john). John is a baseball fan baseball_fan(john). sentence np vp v Hannibal happy is λx.x happyλy.y λx.x happyHannibal Hannibal happy Lambda calculus In the lambda calculus, the semantics of copula be is the identity function, e.g. λy.y Example Derivation: –PhraseLambda Calculus –isλy.y –happyλx.x happy –is happy[λy.y](λx.x happy) – λx.x happy basically the same derivation as... PhraseLambda Calculus barksλx.x barks Shelby barks [λx.x barks](Shelby) Shelby barks

13. sentence npvp v Hannibal is sbar sentence np vp v Shelby saw np e1e1 who 1 More on the Lambda Calculus Back to comparing: –(7) Hannibal is [who Shelby saw] –(7’) Hannibal is happy Semantics (Prolog-style): –(7) Hannibal is [saw(shelby,X)] –(7’) Hannibal is [ happy(X)] Semantics (Lambda calculus): –(7) Hannibal is [λx.Shelby saw x] –(7’) Hannibal is [ λx.x happy] Notice the similarity between (7) and (7’) wrt meaning: –both highlighted parts are single variable λx expressions –(unsaturated for subject) –we can say they are of the “same type” –This means we can use the same identity function λy.y for the copula in either case sentence np vp v Hannibal happy is λx.x happy Hannibal happy (Simplified Derivation) Points to remember: PhraseLambda calculus whoλx ex λy.y Hannibal λy.y saw xShelby [λy.y saw x](Shelby) Shelby saw x λx.Shelby saw x λx λx.Shelby saw x Shelby saw Hannibal x λx.[λy.y saw x]

14. More on the Lambda Calculus We could do topicalization in the same way as for relative clauses 3.4 Topicalization –(9) Shelby, Mary saw –(10) Shelby is who 1 Mary saw e 1 –(10’) Shelby is [λx.Mary saw x] –(10”) Mary saw Shelby